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Theorem onintexmid 4573
Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
onintexmid.onint ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)
Assertion
Ref Expression
onintexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prssi 3751 . . . . . 6 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2 prmg 3714 . . . . . . 7 (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
32adantr 276 . . . . . 6 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4 zfpair2 4211 . . . . . . 7 {𝑢, 𝑣} ∈ V
5 sseq1 3179 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6 eleq2 2241 . . . . . . . . . 10 (𝑦 = {𝑢, 𝑣} → (𝑥𝑦𝑥 ∈ {𝑢, 𝑣}))
76exbidv 1825 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
85, 7anbi12d 473 . . . . . . . 8 (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9 inteq 3848 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
10 id 19 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
119, 10eleq12d 2248 . . . . . . . 8 (𝑦 = {𝑢, 𝑣} → ( 𝑦𝑦 {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
128, 11imbi12d 234 . . . . . . 7 (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13 onintexmid.onint . . . . . . 7 ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)
144, 12, 13vtocl 2792 . . . . . 6 (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})
151, 3, 14syl2anc 411 . . . . 5 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16 elpri 3616 . . . . 5 ( {𝑢, 𝑣} ∈ {𝑢, 𝑣} → ( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣))
1715, 16syl 14 . . . 4 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣))
18 incom 3328 . . . . . . 7 (𝑣𝑢) = (𝑢𝑣)
1918eqeq1i 2185 . . . . . 6 ((𝑣𝑢) = 𝑢 ↔ (𝑢𝑣) = 𝑢)
20 dfss1 3340 . . . . . 6 (𝑢𝑣 ↔ (𝑣𝑢) = 𝑢)
21 vex 2741 . . . . . . . 8 𝑢 ∈ V
22 vex 2741 . . . . . . . 8 𝑣 ∈ V
2321, 22intpr 3877 . . . . . . 7 {𝑢, 𝑣} = (𝑢𝑣)
2423eqeq1i 2185 . . . . . 6 ( {𝑢, 𝑣} = 𝑢 ↔ (𝑢𝑣) = 𝑢)
2519, 20, 243bitr4ri 213 . . . . 5 ( {𝑢, 𝑣} = 𝑢𝑢𝑣)
2623eqeq1i 2185 . . . . . 6 ( {𝑢, 𝑣} = 𝑣 ↔ (𝑢𝑣) = 𝑣)
27 dfss1 3340 . . . . . 6 (𝑣𝑢 ↔ (𝑢𝑣) = 𝑣)
2826, 27bitr4i 187 . . . . 5 ( {𝑢, 𝑣} = 𝑣𝑣𝑢)
2925, 28orbi12i 764 . . . 4 (( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣) ↔ (𝑢𝑣𝑣𝑢))
3017, 29sylib 122 . . 3 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢𝑣𝑣𝑢))
3130rgen2a 2531 . 2 𝑢 ∈ On ∀𝑣 ∈ On (𝑢𝑣𝑣𝑢)
3231ordtri2or2exmid 4571 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708   = wceq 1353  wex 1492  wcel 2148  cin 3129  wss 3130  {cpr 3594   cint 3845  Oncon0 4364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372
This theorem is referenced by: (None)
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